LMFDB - Embedded newform 6036.2.a.g.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6036,2,Mod(1,6036)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6036, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6036 = 2^{2} \cdot 3 \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6036.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$48.1977026600$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 4 x^{14} - 27 x^{13} + 106 x^{12} + 266 x^{11} - 1004 x^{10} - 1105 x^{9} + 4076 x^{8} + \cdots - 44 $ x^15 - 4x^14 - 27x^13 + 106x^12 + 266x^11 - 1004x^10 - 1105x^9 + 4076x^8 + 1501x^7 - 7100x^6 - 134x^5 + 5356x^4 - 1041x^3 - 1381x^2 + 543x - 44
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.61011$ of defining polynomial
Character	$\chi$	$=$	6036.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -3.61011 q^{5} +0.982804 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.61011 q^{5} +0.982804 q^{7} +1.00000 q^{9} -3.24397 q^{11} +2.88731 q^{13} -3.61011 q^{15} -0.659840 q^{17} +2.91239 q^{19} +0.982804 q^{21} -1.79366 q^{23} +8.03288 q^{25} +1.00000 q^{27} +4.94801 q^{29} +1.83768 q^{31} -3.24397 q^{33} -3.54803 q^{35} -11.7471 q^{37} +2.88731 q^{39} -3.21052 q^{41} -7.99865 q^{43} -3.61011 q^{45} +6.45741 q^{47} -6.03410 q^{49} -0.659840 q^{51} +1.47583 q^{53} +11.7111 q^{55} +2.91239 q^{57} -0.809188 q^{59} +9.89425 q^{61} +0.982804 q^{63} -10.4235 q^{65} +7.49893 q^{67} -1.79366 q^{69} -11.1492 q^{71} +10.6862 q^{73} +8.03288 q^{75} -3.18818 q^{77} +5.03225 q^{79} +1.00000 q^{81} +3.32413 q^{83} +2.38209 q^{85} +4.94801 q^{87} -7.58863 q^{89} +2.83766 q^{91} +1.83768 q^{93} -10.5140 q^{95} -2.87712 q^{97} -3.24397 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) $ 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9	$ 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9} - 5 q^{11} - 14 q^{13} - 11 q^{15} - 5 q^{17} + 3 q^{19} - 4 q^{21} - 32 q^{23} + 2 q^{25} + 15 q^{27} - 23 q^{29} - 13 q^{31} - 5 q^{33} - 16 q^{35} - 10 q^{37} - 14 q^{39} - 14 q^{41} + 4 q^{43} - 11 q^{45} - 20 q^{47} - 9 q^{49} - 5 q^{51} - 30 q^{53} - 10 q^{55} + 3 q^{57} - 14 q^{59} - 38 q^{61} - 4 q^{63} - 24 q^{65} - 8 q^{67} - 32 q^{69} - 41 q^{71} - 19 q^{73} + 2 q^{75} - 39 q^{77} - 27 q^{79} + 15 q^{81} - 17 q^{83} - 6 q^{85} - 23 q^{87} - 23 q^{89} + 4 q^{91} - 13 q^{93} - 30 q^{95} - 18 q^{97} - 5 q^{99}+O(q^{100}) $ 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9 - 5 * q^11 - 14 * q^13 - 11 * q^15 - 5 * q^17 + 3 * q^19 - 4 * q^21 - 32 * q^23 + 2 * q^25 + 15 * q^27 - 23 * q^29 - 13 * q^31 - 5 * q^33 - 16 * q^35 - 10 * q^37 - 14 * q^39 - 14 * q^41 + 4 * q^43 - 11 * q^45 - 20 * q^47 - 9 * q^49 - 5 * q^51 - 30 * q^53 - 10 * q^55 + 3 * q^57 - 14 * q^59 - 38 * q^61 - 4 * q^63 - 24 * q^65 - 8 * q^67 - 32 * q^69 - 41 * q^71 - 19 * q^73 + 2 * q^75 - 39 * q^77 - 27 * q^79 + 15 * q^81 - 17 * q^83 - 6 * q^85 - 23 * q^87 - 23 * q^89 + 4 * q^91 - 13 * q^93 - 30 * q^95 - 18 * q^97 - 5 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−3.61011	−1.61449	−0.807245	−	0.590217i	$-0.799042\pi$
$5$	−3.61011	−1.61449	−0.807245	+	0.590217i	$0.799042\pi$
$6$	0	0
$7$	0.982804	0.371465	0.185732	−	0.982600i	$-0.440534\pi$
$7$	0.982804	0.371465	0.185732	+	0.982600i	$0.440534\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.24397	−0.978093	−0.489047	−	0.872258i	$-0.662655\pi$
$11$	−3.24397	−0.978093	−0.489047	+	0.872258i	$0.662655\pi$
$12$	0	0
$13$	2.88731	0.800797	0.400398	−	0.916341i	$-0.368872\pi$
$13$	2.88731	0.800797	0.400398	+	0.916341i	$0.368872\pi$
$14$	0	0
$15$	−3.61011	−0.932126
$16$	0	0
$17$	−0.659840	−0.160035	−0.0800173	−	0.996793i	$-0.525498\pi$
$17$	−0.659840	−0.160035	−0.0800173	+	0.996793i	$0.525498\pi$
$18$	0	0
$19$	2.91239	0.668147	0.334074	−	0.942547i	$-0.391577\pi$
$19$	2.91239	0.668147	0.334074	+	0.942547i	$0.391577\pi$
$20$	0	0
$21$	0.982804	0.214465
$22$	0	0
$23$	−1.79366	−0.374003	−0.187002	−	0.982360i	$-0.559877\pi$
$23$	−1.79366	−0.374003	−0.187002	+	0.982360i	$0.559877\pi$
$24$	0	0
$25$	8.03288	1.60658
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	4.94801	0.918822	0.459411	−	0.888224i	$-0.348061\pi$
$29$	4.94801	0.918822	0.459411	+	0.888224i	$0.348061\pi$
$30$	0	0
$31$	1.83768	0.330057	0.165029	−	0.986289i	$-0.447228\pi$
$31$	1.83768	0.330057	0.165029	+	0.986289i	$0.447228\pi$
$32$	0	0
$33$	−3.24397	−0.564702
$34$	0	0
$35$	−3.54803	−0.599726
$36$	0	0
$37$	−11.7471	−1.93122	−0.965609	−	0.260000i	$-0.916277\pi$
$37$	−11.7471	−1.93122	−0.965609	+	0.260000i	$0.916277\pi$
$38$	0	0
$39$	2.88731	0.462340
$40$	0	0
$41$	−3.21052	−0.501398	−0.250699	−	0.968065i	$-0.580661\pi$
$41$	−3.21052	−0.501398	−0.250699	+	0.968065i	$0.580661\pi$
$42$	0	0
$43$	−7.99865	−1.21978	−0.609891	−	0.792485i	$-0.708787\pi$
$43$	−7.99865	−1.21978	−0.609891	+	0.792485i	$0.708787\pi$
$44$	0	0
$45$	−3.61011	−0.538163
$46$	0	0
$47$	6.45741	0.941910	0.470955	−	0.882157i	$-0.343910\pi$
$47$	6.45741	0.941910	0.470955	+	0.882157i	$0.343910\pi$
$48$	0	0
$49$	−6.03410	−0.862014
$50$	0	0
$51$	−0.659840	−0.0923960
$52$	0	0
$53$	1.47583	0.202721	0.101360	−	0.994850i	$-0.467681\pi$
$53$	1.47583	0.202721	0.101360	+	0.994850i	$0.467681\pi$
$54$	0	0
$55$	11.7111	1.57912
$56$	0	0
$57$	2.91239	0.385755
$58$	0	0
$59$	−0.809188	−0.105347	−0.0526737	−	0.998612i	$-0.516774\pi$
$59$	−0.809188	−0.105347	−0.0526737	+	0.998612i	$0.516774\pi$
$60$	0	0
$61$	9.89425	1.26683	0.633414	−	0.773813i	$-0.281653\pi$
$61$	9.89425	1.26683	0.633414	+	0.773813i	$0.281653\pi$
$62$	0	0
$63$	0.982804	0.123822
$64$	0	0
$65$	−10.4235	−1.29288
$66$	0	0
$67$	7.49893	0.916140	0.458070	−	0.888916i	$-0.348541\pi$
$67$	7.49893	0.916140	0.458070	+	0.888916i	$0.348541\pi$
$68$	0	0
$69$	−1.79366	−0.215931
$70$	0	0
$71$	−11.1492	−1.32316	−0.661582	−	0.749873i	$-0.730115\pi$
$71$	−11.1492	−1.32316	−0.661582	+	0.749873i	$0.730115\pi$
$72$	0	0
$73$	10.6862	1.25072	0.625362	−	0.780335i	$-0.284951\pi$
$73$	10.6862	1.25072	0.625362	+	0.780335i	$0.284951\pi$
$74$	0	0
$75$	8.03288	0.927557
$76$	0	0
$77$	−3.18818	−0.363327
$78$	0	0
$79$	5.03225	0.566172	0.283086	−	0.959094i	$-0.408642\pi$
$79$	5.03225	0.566172	0.283086	+	0.959094i	$0.408642\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.32413	0.364871	0.182435	−	0.983218i	$-0.441602\pi$
$83$	3.32413	0.364871	0.182435	+	0.983218i	$0.441602\pi$
$84$	0	0
$85$	2.38209	0.258374
$86$	0	0
$87$	4.94801	0.530482
$88$	0	0
$89$	−7.58863	−0.804394	−0.402197	−	0.915553i	$-0.631753\pi$
$89$	−7.58863	−0.804394	−0.402197	+	0.915553i	$0.631753\pi$
$90$	0	0
$91$	2.83766	0.297468
$92$	0	0
$93$	1.83768	0.190559
$94$	0	0
$95$	−10.5140	−1.07872
$96$	0	0
$97$	−2.87712	−0.292127	−0.146064	−	0.989275i	$-0.546660\pi$
$97$	−2.87712	−0.292127	−0.146064	+	0.989275i	$0.546660\pi$
$98$	0	0
$99$	−3.24397	−0.326031
$100$	0	0
$101$	−16.3446	−1.62635	−0.813174	−	0.582021i	$-0.802262\pi$
$101$	−16.3446	−1.62635	−0.813174	+	0.582021i	$0.802262\pi$
$102$	0	0
$103$	2.54509	0.250775	0.125388	−	0.992108i	$-0.459983\pi$
$103$	2.54509	0.250775	0.125388	+	0.992108i	$0.459983\pi$
$104$	0	0
$105$	−3.54803	−0.346252
$106$	0	0
$107$	−4.87320	−0.471110	−0.235555	−	0.971861i	$-0.575691\pi$
$107$	−4.87320	−0.471110	−0.235555	+	0.971861i	$0.575691\pi$
$108$	0	0
$109$	7.63845	0.731631	0.365816	−	0.930687i	$-0.380790\pi$
$109$	7.63845	0.731631	0.365816	+	0.930687i	$0.380790\pi$
$110$	0	0
$111$	−11.7471	−1.11499
$112$	0	0
$113$	−5.11980	−0.481630	−0.240815	−	0.970571i	$-0.577415\pi$
$113$	−5.11980	−0.481630	−0.240815	+	0.970571i	$0.577415\pi$
$114$	0	0
$115$	6.47529	0.603824
$116$	0	0
$117$	2.88731	0.266932
$118$	0	0
$119$	−0.648493	−0.0594472
$120$	0	0
$121$	−0.476671	−0.0433337
$122$	0	0
$123$	−3.21052	−0.289483
$124$	0	0
$125$	−10.9490	−0.979309
$126$	0	0
$127$	−17.6475	−1.56596	−0.782979	−	0.622048i	$-0.786301\pi$
$127$	−17.6475	−1.56596	−0.782979	+	0.622048i	$0.786301\pi$
$128$	0	0
$129$	−7.99865	−0.704242
$130$	0	0
$131$	1.70788	0.149218	0.0746090	−	0.997213i	$-0.476229\pi$
$131$	1.70788	0.149218	0.0746090	+	0.997213i	$0.476229\pi$
$132$	0	0
$133$	2.86231	0.248193
$134$	0	0
$135$	−3.61011	−0.310709
$136$	0	0
$137$	−18.5861	−1.58792	−0.793959	−	0.607971i	$-0.791984\pi$
$137$	−18.5861	−1.58792	−0.793959	+	0.607971i	$0.791984\pi$
$138$	0	0
$139$	3.89774	0.330602	0.165301	−	0.986243i	$-0.447140\pi$
$139$	3.89774	0.330602	0.165301	+	0.986243i	$0.447140\pi$
$140$	0	0
$141$	6.45741	0.543812
$142$	0	0
$143$	−9.36636	−0.783254
$144$	0	0
$145$	−17.8628	−1.48343
$146$	0	0
$147$	−6.03410	−0.497684
$148$	0	0
$149$	4.10505	0.336299	0.168149	−	0.985762i	$-0.446221\pi$
$149$	4.10505	0.336299	0.168149	+	0.985762i	$0.446221\pi$
$150$	0	0
$151$	−14.4204	−1.17351	−0.586757	−	0.809763i	$-0.699596\pi$
$151$	−14.4204	−1.17351	−0.586757	+	0.809763i	$0.699596\pi$
$152$	0	0
$153$	−0.659840	−0.0533449
$154$	0	0
$155$	−6.63422	−0.532874
$156$	0	0
$157$	4.50079	0.359202	0.179601	−	0.983740i	$-0.442519\pi$
$157$	4.50079	0.359202	0.179601	+	0.983740i	$0.442519\pi$
$158$	0	0
$159$	1.47583	0.117041
$160$	0	0
$161$	−1.76281	−0.138929
$162$	0	0
$163$	13.2204	1.03550	0.517749	−	0.855532i	$-0.326770\pi$
$163$	13.2204	1.03550	0.517749	+	0.855532i	$0.326770\pi$
$164$	0	0
$165$	11.7111	0.911706
$166$	0	0
$167$	−24.6872	−1.91035	−0.955176	−	0.296038i	$-0.904334\pi$
$167$	−24.6872	−1.91035	−0.955176	+	0.296038i	$0.904334\pi$
$168$	0	0
$169$	−4.66342	−0.358724
$170$	0	0
$171$	2.91239	0.222716
$172$	0	0
$173$	−15.0619	−1.14513	−0.572566	−	0.819859i	$-0.694052\pi$
$173$	−15.0619	−1.14513	−0.572566	+	0.819859i	$0.694052\pi$
$174$	0	0
$175$	7.89474	0.596786
$176$	0	0
$177$	−0.809188	−0.0608223
$178$	0	0
$179$	−21.9460	−1.64032	−0.820161	−	0.572134i	$-0.806116\pi$
$179$	−21.9460	−1.64032	−0.820161	+	0.572134i	$0.806116\pi$
$180$	0	0
$181$	−20.3387	−1.51176	−0.755882	−	0.654708i	$-0.772792\pi$
$181$	−20.3387	−1.51176	−0.755882	+	0.654708i	$0.772792\pi$
$182$	0	0
$183$	9.89425	0.731404
$184$	0	0
$185$	42.4084	3.11793
$186$	0	0
$187$	2.14050	0.156529
$188$	0	0
$189$	0.982804	0.0714885
$190$	0	0
$191$	5.04043	0.364713	0.182356	−	0.983233i	$-0.441628\pi$
$191$	5.04043	0.364713	0.182356	+	0.983233i	$0.441628\pi$
$192$	0	0
$193$	1.77910	0.128062	0.0640311	−	0.997948i	$-0.479604\pi$
$193$	1.77910	0.128062	0.0640311	+	0.997948i	$0.479604\pi$
$194$	0	0
$195$	−10.4235	−0.746443
$196$	0	0
$197$	3.17972	0.226546	0.113273	−	0.993564i	$-0.463867\pi$
$197$	3.17972	0.226546	0.113273	+	0.993564i	$0.463867\pi$
$198$	0	0
$199$	−12.0516	−0.854312	−0.427156	−	0.904178i	$-0.640485\pi$
$199$	−12.0516	−0.854312	−0.427156	+	0.904178i	$0.640485\pi$
$200$	0	0
$201$	7.49893	0.528934
$202$	0	0
$203$	4.86292	0.341310
$204$	0	0
$205$	11.5903	0.809502
$206$	0	0
$207$	−1.79366	−0.124668
$208$	0	0
$209$	−9.44769	−0.653510
$210$	0	0
$211$	−6.06555	−0.417570	−0.208785	−	0.977962i	$-0.566951\pi$
$211$	−6.06555	−0.417570	−0.208785	+	0.977962i	$0.566951\pi$
$212$	0	0
$213$	−11.1492	−0.763929
$214$	0	0
$215$	28.8760	1.96933
$216$	0	0
$217$	1.80608	0.122605
$218$	0	0
$219$	10.6862	0.722105
$220$	0	0
$221$	−1.90516	−0.128155
$222$	0	0
$223$	−27.4945	−1.84117	−0.920583	−	0.390547i	$-0.872286\pi$
$223$	−27.4945	−1.84117	−0.920583	+	0.390547i	$0.872286\pi$
$224$	0	0
$225$	8.03288	0.535525
$226$	0	0
$227$	−0.283813	−0.0188373	−0.00941867	−	0.999956i	$-0.502998\pi$
$227$	−0.283813	−0.0188373	−0.00941867	+	0.999956i	$0.502998\pi$
$228$	0	0
$229$	17.0508	1.12675	0.563375	−	0.826201i	$-0.309503\pi$
$229$	17.0508	1.12675	0.563375	+	0.826201i	$0.309503\pi$
$230$	0	0
$231$	−3.18818	−0.209767
$232$	0	0
$233$	2.24231	0.146899	0.0734494	−	0.997299i	$-0.476599\pi$
$233$	2.24231	0.146899	0.0734494	+	0.997299i	$0.476599\pi$
$234$	0	0
$235$	−23.3119	−1.52070
$236$	0	0
$237$	5.03225	0.326880
$238$	0	0
$239$	−3.40186	−0.220048	−0.110024	−	0.993929i	$-0.535093\pi$
$239$	−3.40186	−0.220048	−0.110024	+	0.993929i	$0.535093\pi$
$240$	0	0
$241$	−17.0872	−1.10069	−0.550343	−	0.834939i	$-0.685503\pi$
$241$	−17.0872	−1.10069	−0.550343	+	0.834939i	$0.685503\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	21.7837	1.39171
$246$	0	0
$247$	8.40898	0.535050
$248$	0	0
$249$	3.32413	0.210658
$250$	0	0
$251$	30.1261	1.90154	0.950771	−	0.309894i	$-0.100293\pi$
$251$	30.1261	1.90154	0.950771	+	0.309894i	$0.100293\pi$
$252$	0	0
$253$	5.81857	0.365810
$254$	0	0
$255$	2.38209	0.149172
$256$	0	0
$257$	−12.5832	−0.784919	−0.392459	−	0.919769i	$-0.628376\pi$
$257$	−12.5832	−0.784919	−0.392459	+	0.919769i	$0.628376\pi$
$258$	0	0
$259$	−11.5451	−0.717380
$260$	0	0
$261$	4.94801	0.306274
$262$	0	0
$263$	−30.6752	−1.89151	−0.945755	−	0.324879i	$-0.894676\pi$
$263$	−30.6752	−1.89151	−0.945755	+	0.324879i	$0.894676\pi$
$264$	0	0
$265$	−5.32790	−0.327290
$266$	0	0
$267$	−7.58863	−0.464417
$268$	0	0
$269$	−19.8127	−1.20800	−0.604002	−	0.796983i	$-0.706428\pi$
$269$	−19.8127	−1.20800	−0.604002	+	0.796983i	$0.706428\pi$
$270$	0	0
$271$	−4.89053	−0.297078	−0.148539	−	0.988907i	$-0.547457\pi$
$271$	−4.89053	−0.297078	−0.148539	+	0.988907i	$0.547457\pi$
$272$	0	0
$273$	2.83766	0.171743
$274$	0	0
$275$	−26.0584	−1.57138
$276$	0	0
$277$	4.15322	0.249543	0.124771	−	0.992186i	$-0.460180\pi$
$277$	4.15322	0.249543	0.124771	+	0.992186i	$0.460180\pi$
$278$	0	0
$279$	1.83768	0.110019
$280$	0	0
$281$	31.7832	1.89603	0.948014	−	0.318229i	$-0.103088\pi$
$281$	31.7832	1.89603	0.948014	+	0.318229i	$0.103088\pi$
$282$	0	0
$283$	11.6889	0.694832	0.347416	−	0.937711i	$-0.387059\pi$
$283$	11.6889	0.694832	0.347416	+	0.937711i	$0.387059\pi$
$284$	0	0
$285$	−10.5140	−0.622797
$286$	0	0
$287$	−3.15531	−0.186252
$288$	0	0
$289$	−16.5646	−0.974389
$290$	0	0
$291$	−2.87712	−0.168660
$292$	0	0
$293$	29.6797	1.73391	0.866954	−	0.498389i	$-0.166075\pi$
$293$	29.6797	1.73391	0.866954	+	0.498389i	$0.166075\pi$
$294$	0	0
$295$	2.92126	0.170082
$296$	0	0
$297$	−3.24397	−0.188234
$298$	0	0
$299$	−5.17885	−0.299501
$300$	0	0
$301$	−7.86110	−0.453106
$302$	0	0
$303$	−16.3446	−0.938972
$304$	0	0
$305$	−35.7193	−2.04528
$306$	0	0
$307$	8.25970	0.471406	0.235703	−	0.971825i	$-0.424261\pi$
$307$	8.25970	0.471406	0.235703	+	0.971825i	$0.424261\pi$
$308$	0	0
$309$	2.54509	0.144785
$310$	0	0
$311$	−22.6765	−1.28587	−0.642934	−	0.765921i	$-0.722283\pi$
$311$	−22.6765	−1.28587	−0.642934	+	0.765921i	$0.722283\pi$
$312$	0	0
$313$	18.6755	1.05560	0.527800	−	0.849369i	$-0.323017\pi$
$313$	18.6755	1.05560	0.527800	+	0.849369i	$0.323017\pi$
$314$	0	0
$315$	−3.54803	−0.199909
$316$	0	0
$317$	−10.1281	−0.568849	−0.284425	−	0.958698i	$-0.591803\pi$
$317$	−10.1281	−0.568849	−0.284425	+	0.958698i	$0.591803\pi$
$318$	0	0
$319$	−16.0512	−0.898693
$320$	0	0
$321$	−4.87320	−0.271995
$322$	0	0
$323$	−1.92171	−0.106927
$324$	0	0
$325$	23.1934	1.28654
$326$	0	0
$327$	7.63845	0.422407
$328$	0	0
$329$	6.34636	0.349886
$330$	0	0
$331$	−26.0075	−1.42950	−0.714752	−	0.699378i	$-0.753460\pi$
$331$	−26.0075	−1.42950	−0.714752	+	0.699378i	$0.753460\pi$
$332$	0	0
$333$	−11.7471	−0.643739
$334$	0	0
$335$	−27.0719	−1.47910
$336$	0	0
$337$	−5.94577	−0.323887	−0.161943	−	0.986800i	$-0.551776\pi$
$337$	−5.94577	−0.323887	−0.161943	+	0.986800i	$0.551776\pi$
$338$	0	0
$339$	−5.11980	−0.278069
$340$	0	0
$341$	−5.96138	−0.322827
$342$	0	0
$343$	−12.8100	−0.691673
$344$	0	0
$345$	6.47529	0.348618
$346$	0	0
$347$	−17.2019	−0.923446	−0.461723	−	0.887024i	$-0.652769\pi$
$347$	−17.2019	−0.923446	−0.461723	+	0.887024i	$0.652769\pi$
$348$	0	0
$349$	−24.9725	−1.33675	−0.668373	−	0.743826i	$-0.733009\pi$
$349$	−24.9725	−1.33675	−0.668373	+	0.743826i	$0.733009\pi$
$350$	0	0
$351$	2.88731	0.154113
$352$	0	0
$353$	19.7066	1.04888	0.524439	−	0.851448i	$-0.324275\pi$
$353$	19.7066	1.04888	0.524439	+	0.851448i	$0.324275\pi$
$354$	0	0
$355$	40.2497	2.13623
$356$	0	0
$357$	−0.648493	−0.0343219
$358$	0	0
$359$	−28.4459	−1.50132	−0.750659	−	0.660690i	$-0.770264\pi$
$359$	−28.4459	−1.50132	−0.750659	+	0.660690i	$0.770264\pi$
$360$	0	0
$361$	−10.5180	−0.553579
$362$	0	0
$363$	−0.476671	−0.0250187
$364$	0	0
$365$	−38.5783	−2.01928
$366$	0	0
$367$	−6.27326	−0.327461	−0.163731	−	0.986505i	$-0.552353\pi$
$367$	−6.27326	−0.327461	−0.163731	+	0.986505i	$0.552353\pi$
$368$	0	0
$369$	−3.21052	−0.167133
$370$	0	0
$371$	1.45045	0.0753036
$372$	0	0
$373$	−12.2395	−0.633739	−0.316869	−	0.948469i	$-0.602632\pi$
$373$	−12.2395	−0.633739	−0.316869	+	0.948469i	$0.602632\pi$
$374$	0	0
$375$	−10.9490	−0.565404
$376$	0	0
$377$	14.2865	0.735790
$378$	0	0
$379$	35.1898	1.80758	0.903791	−	0.427975i	$-0.140773\pi$
$379$	35.1898	1.80758	0.903791	+	0.427975i	$0.140773\pi$
$380$	0	0
$381$	−17.6475	−0.904107
$382$	0	0
$383$	−17.5142	−0.894933	−0.447466	−	0.894301i	$-0.647674\pi$
$383$	−17.5142	−0.894933	−0.447466	+	0.894301i	$0.647674\pi$
$384$	0	0
$385$	11.5097	0.586588
$386$	0	0
$387$	−7.99865	−0.406594
$388$	0	0
$389$	−20.0155	−1.01482	−0.507412	−	0.861703i	$-0.669398\pi$
$389$	−20.0155	−1.01482	−0.507412	+	0.861703i	$0.669398\pi$
$390$	0	0
$391$	1.18353	0.0598535
$392$	0	0
$393$	1.70788	0.0861511
$394$	0	0
$395$	−18.1670	−0.914079
$396$	0	0
$397$	10.1966	0.511754	0.255877	−	0.966709i	$-0.417636\pi$
$397$	10.1966	0.511754	0.255877	+	0.966709i	$0.417636\pi$
$398$	0	0
$399$	2.86231	0.143294
$400$	0	0
$401$	35.4700	1.77129	0.885644	−	0.464365i	$-0.153718\pi$
$401$	35.4700	1.77129	0.885644	+	0.464365i	$0.153718\pi$
$402$	0	0
$403$	5.30596	0.264309
$404$	0	0
$405$	−3.61011	−0.179388
$406$	0	0
$407$	38.1073	1.88891
$408$	0	0
$409$	0.916877	0.0453367	0.0226683	−	0.999743i	$-0.492784\pi$
$409$	0.916877	0.0453367	0.0226683	+	0.999743i	$0.492784\pi$
$410$	0	0
$411$	−18.5861	−0.916785
$412$	0	0
$413$	−0.795273	−0.0391328
$414$	0	0
$415$	−12.0005	−0.589080
$416$	0	0
$417$	3.89774	0.190873
$418$	0	0
$419$	12.3084	0.601306	0.300653	−	0.953734i	$-0.402795\pi$
$419$	12.3084	0.601306	0.300653	+	0.953734i	$0.402795\pi$
$420$	0	0
$421$	−19.6876	−0.959513	−0.479756	−	0.877402i	$-0.659275\pi$
$421$	−19.6876	−0.959513	−0.479756	+	0.877402i	$0.659275\pi$
$422$	0	0
$423$	6.45741	0.313970
$424$	0	0
$425$	−5.30041	−0.257108
$426$	0	0
$427$	9.72410	0.470582
$428$	0	0
$429$	−9.36636	−0.452212
$430$	0	0
$431$	−13.5878	−0.654502	−0.327251	−	0.944937i	$-0.606122\pi$
$431$	−13.5878	−0.654502	−0.327251	+	0.944937i	$0.606122\pi$
$432$	0	0
$433$	31.3972	1.50886	0.754428	−	0.656383i	$-0.227915\pi$
$433$	31.3972	1.50886	0.754428	+	0.656383i	$0.227915\pi$
$434$	0	0
$435$	−17.8628	−0.856458
$436$	0	0
$437$	−5.22382	−0.249889
$438$	0	0
$439$	39.8580	1.90232	0.951159	−	0.308702i	$-0.0998946\pi$
$439$	39.8580	1.90232	0.951159	+	0.308702i	$0.0998946\pi$
$440$	0	0
$441$	−6.03410	−0.287338
$442$	0	0
$443$	10.2583	0.487388	0.243694	−	0.969852i	$-0.421641\pi$
$443$	10.2583	0.487388	0.243694	+	0.969852i	$0.421641\pi$
$444$	0	0
$445$	27.3958	1.29868
$446$	0	0
$447$	4.10505	0.194162
$448$	0	0
$449$	16.6778	0.787076	0.393538	−	0.919308i	$-0.371251\pi$
$449$	16.6778	0.787076	0.393538	+	0.919308i	$0.371251\pi$
$450$	0	0
$451$	10.4148	0.490414
$452$	0	0
$453$	−14.4204	−0.677529
$454$	0	0
$455$	−10.2443	−0.480259
$456$	0	0
$457$	−5.16028	−0.241388	−0.120694	−	0.992690i	$-0.538512\pi$
$457$	−5.16028	−0.241388	−0.120694	+	0.992690i	$0.538512\pi$
$458$	0	0
$459$	−0.659840	−0.0307987
$460$	0	0
$461$	26.8986	1.25279	0.626397	−	0.779504i	$-0.284529\pi$
$461$	26.8986	1.25279	0.626397	+	0.779504i	$0.284529\pi$
$462$	0	0
$463$	19.9121	0.925394	0.462697	−	0.886516i	$-0.346882\pi$
$463$	19.9121	0.925394	0.462697	+	0.886516i	$0.346882\pi$
$464$	0	0
$465$	−6.63422	−0.307655
$466$	0	0
$467$	−18.8936	−0.874291	−0.437145	−	0.899391i	$-0.644010\pi$
$467$	−18.8936	−0.874291	−0.437145	+	0.899391i	$0.644010\pi$
$468$	0	0
$469$	7.36998	0.340314
$470$	0	0
$471$	4.50079	0.207385
$472$	0	0
$473$	25.9474	1.19306
$474$	0	0
$475$	23.3948	1.07343
$476$	0	0
$477$	1.47583	0.0675735
$478$	0	0
$479$	16.4848	0.753208	0.376604	−	0.926374i	$-0.377092\pi$
$479$	16.4848	0.753208	0.376604	+	0.926374i	$0.377092\pi$
$480$	0	0
$481$	−33.9177	−1.54651
$482$	0	0
$483$	−1.76281	−0.0802108
$484$	0	0
$485$	10.3867	0.471636
$486$	0	0
$487$	8.18003	0.370672	0.185336	−	0.982675i	$-0.440663\pi$
$487$	8.18003	0.370672	0.185336	+	0.982675i	$0.440663\pi$
$488$	0	0
$489$	13.2204	0.597845
$490$	0	0
$491$	−6.85552	−0.309385	−0.154693	−	0.987963i	$-0.549439\pi$
$491$	−6.85552	−0.309385	−0.154693	+	0.987963i	$0.549439\pi$
$492$	0	0
$493$	−3.26489	−0.147043
$494$	0	0
$495$	11.7111	0.526374
$496$	0	0
$497$	−10.9575	−0.491509
$498$	0	0
$499$	14.7738	0.661366	0.330683	−	0.943742i	$-0.392721\pi$
$499$	14.7738	0.661366	0.330683	+	0.943742i	$0.392721\pi$
$500$	0	0
$501$	−24.6872	−1.10294
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	59.0057	2.62572
$506$	0	0
$507$	−4.66342	−0.207110
$508$	0	0
$509$	−17.5415	−0.777513	−0.388756	−	0.921341i	$-0.627095\pi$
$509$	−17.5415	−0.777513	−0.388756	+	0.921341i	$0.627095\pi$
$510$	0	0
$511$	10.5024	0.464600
$512$	0	0
$513$	2.91239	0.128585
$514$	0	0
$515$	−9.18806	−0.404874
$516$	0	0
$517$	−20.9476	−0.921275
$518$	0	0
$519$	−15.0619	−0.661142
$520$	0	0
$521$	15.1933	0.665629	0.332815	−	0.942992i	$-0.392002\pi$
$521$	15.1933	0.665629	0.332815	+	0.942992i	$0.392002\pi$
$522$	0	0
$523$	11.1425	0.487227	0.243613	−	0.969872i	$-0.421667\pi$
$523$	11.1425	0.487227	0.243613	+	0.969872i	$0.421667\pi$
$524$	0	0
$525$	7.89474	0.344555
$526$	0	0
$527$	−1.21257	−0.0528205
$528$	0	0
$529$	−19.7828	−0.860122
$530$	0	0
$531$	−0.809188	−0.0351158
$532$	0	0
$533$	−9.26977	−0.401518
$534$	0	0
$535$	17.5928	0.760602
$536$	0	0
$537$	−21.9460	−0.947040
$538$	0	0
$539$	19.5744	0.843130
$540$	0	0
$541$	−37.2768	−1.60265	−0.801327	−	0.598227i	$-0.795872\pi$
$541$	−37.2768	−1.60265	−0.801327	+	0.598227i	$0.795872\pi$
$542$	0	0
$543$	−20.3387	−0.872817
$544$	0	0
$545$	−27.5756	−1.18121
$546$	0	0
$547$	−4.50714	−0.192711	−0.0963556	−	0.995347i	$-0.530719\pi$
$547$	−4.50714	−0.192711	−0.0963556	+	0.995347i	$0.530719\pi$
$548$	0	0
$549$	9.89425	0.422276
$550$	0	0
$551$	14.4105	0.613909
$552$	0	0
$553$	4.94571	0.210313
$554$	0	0
$555$	42.4084	1.80014
$556$	0	0
$557$	8.53654	0.361705	0.180852	−	0.983510i	$-0.442114\pi$
$557$	8.53654	0.361705	0.180852	+	0.983510i	$0.442114\pi$
$558$	0	0
$559$	−23.0946	−0.976798
$560$	0	0
$561$	2.14050	0.0903719
$562$	0	0
$563$	−9.17805	−0.386809	−0.193404	−	0.981119i	$-0.561953\pi$
$563$	−9.17805	−0.386809	−0.193404	+	0.981119i	$0.561953\pi$
$564$	0	0
$565$	18.4830	0.777586
$566$	0	0
$567$	0.982804	0.0412739
$568$	0	0
$569$	4.61067	0.193289	0.0966447	−	0.995319i	$-0.469189\pi$
$569$	4.61067	0.193289	0.0966447	+	0.995319i	$0.469189\pi$
$570$	0	0
$571$	−42.0793	−1.76097	−0.880483	−	0.474078i	$-0.842781\pi$
$571$	−42.0793	−1.76097	−0.880483	+	0.474078i	$0.842781\pi$
$572$	0	0
$573$	5.04043	0.210567
$574$	0	0
$575$	−14.4082	−0.600864
$576$	0	0
$577$	−18.7259	−0.779568	−0.389784	−	0.920906i	$-0.627450\pi$
$577$	−18.7259	−0.779568	−0.389784	+	0.920906i	$0.627450\pi$
$578$	0	0
$579$	1.77910	0.0739368
$580$	0	0
$581$	3.26697	0.135537
$582$	0	0
$583$	−4.78754	−0.198280
$584$	0	0
$585$	−10.4235	−0.430959
$586$	0	0
$587$	17.3897	0.717751	0.358875	−	0.933386i	$-0.383160\pi$
$587$	17.3897	0.717751	0.358875	+	0.933386i	$0.383160\pi$
$588$	0	0
$589$	5.35204	0.220527
$590$	0	0
$591$	3.17972	0.130796
$592$	0	0
$593$	−12.8250	−0.526659	−0.263329	−	0.964706i	$-0.584821\pi$
$593$	−12.8250	−0.526659	−0.263329	+	0.964706i	$0.584821\pi$
$594$	0	0
$595$	2.34113	0.0959769
$596$	0	0
$597$	−12.0516	−0.493238
$598$	0	0
$599$	39.8287	1.62736	0.813679	−	0.581314i	$-0.197461\pi$
$599$	39.8287	1.62736	0.813679	+	0.581314i	$0.197461\pi$
$600$	0	0
$601$	−14.5019	−0.591544	−0.295772	−	0.955259i	$-0.595577\pi$
$601$	−14.5019	−0.591544	−0.295772	+	0.955259i	$0.595577\pi$
$602$	0	0
$603$	7.49893	0.305380
$604$	0	0
$605$	1.72083	0.0699619
$606$	0	0
$607$	40.7344	1.65336	0.826679	−	0.562674i	$-0.190227\pi$
$607$	40.7344	1.65336	0.826679	+	0.562674i	$0.190227\pi$
$608$	0	0
$609$	4.86292	0.197055
$610$	0	0
$611$	18.6446	0.754278
$612$	0	0
$613$	10.7028	0.432283	0.216142	−	0.976362i	$-0.430653\pi$
$613$	10.7028	0.432283	0.216142	+	0.976362i	$0.430653\pi$
$614$	0	0
$615$	11.5903	0.467366
$616$	0	0
$617$	16.7939	0.676097	0.338048	−	0.941129i	$-0.390233\pi$
$617$	16.7939	0.676097	0.338048	+	0.941129i	$0.390233\pi$
$618$	0	0
$619$	16.2650	0.653747	0.326874	−	0.945068i	$-0.394005\pi$
$619$	16.2650	0.653747	0.326874	+	0.945068i	$0.394005\pi$
$620$	0	0
$621$	−1.79366	−0.0719770
$622$	0	0
$623$	−7.45814	−0.298804
$624$	0	0
$625$	−0.637277	−0.0254911
$626$	0	0
$627$	−9.44769	−0.377304
$628$	0	0
$629$	7.75122	0.309062
$630$	0	0
$631$	−4.68508	−0.186510	−0.0932550	−	0.995642i	$-0.529727\pi$
$631$	−4.68508	−0.186510	−0.0932550	+	0.995642i	$0.529727\pi$
$632$	0	0
$633$	−6.06555	−0.241084
$634$	0	0
$635$	63.7092	2.52822
$636$	0	0
$637$	−17.4223	−0.690298
$638$	0	0
$639$	−11.1492	−0.441055
$640$	0	0
$641$	28.2973	1.11768	0.558838	−	0.829277i	$-0.311247\pi$
$641$	28.2973	1.11768	0.558838	+	0.829277i	$0.311247\pi$
$642$	0	0
$643$	−10.0424	−0.396034	−0.198017	−	0.980199i	$-0.563450\pi$
$643$	−10.0424	−0.396034	−0.198017	+	0.980199i	$0.563450\pi$
$644$	0	0
$645$	28.8760	1.13699
$646$	0	0
$647$	−1.03715	−0.0407745	−0.0203872	−	0.999792i	$-0.506490\pi$
$647$	−1.03715	−0.0407745	−0.0203872	+	0.999792i	$0.506490\pi$
$648$	0	0
$649$	2.62498	0.103040
$650$	0	0
$651$	1.80608	0.0707858
$652$	0	0
$653$	13.4029	0.524497	0.262249	−	0.965000i	$-0.415536\pi$
$653$	13.4029	0.524497	0.262249	+	0.965000i	$0.415536\pi$
$654$	0	0
$655$	−6.16562	−0.240911
$656$	0	0
$657$	10.6862	0.416908
$658$	0	0
$659$	24.5472	0.956222	0.478111	−	0.878300i	$-0.341322\pi$
$659$	24.5472	0.956222	0.478111	+	0.878300i	$0.341322\pi$
$660$	0	0
$661$	−20.5544	−0.799474	−0.399737	−	0.916630i	$-0.630899\pi$
$661$	−20.5544	−0.799474	−0.399737	+	0.916630i	$0.630899\pi$
$662$	0	0
$663$	−1.90516	−0.0739904
$664$	0	0
$665$	−10.3332	−0.400705
$666$	0	0
$667$	−8.87503	−0.343642
$668$	0	0
$669$	−27.4945	−1.06300
$670$	0	0
$671$	−32.0966	−1.23908
$672$	0	0
$673$	−26.8226	−1.03394	−0.516968	−	0.856004i	$-0.672940\pi$
$673$	−26.8226	−1.03394	−0.516968	+	0.856004i	$0.672940\pi$
$674$	0	0
$675$	8.03288	0.309186
$676$	0	0
$677$	−14.8612	−0.571164	−0.285582	−	0.958354i	$-0.592187\pi$
$677$	−14.8612	−0.571164	−0.285582	+	0.958354i	$0.592187\pi$
$678$	0	0
$679$	−2.82764	−0.108515
$680$	0	0
$681$	−0.283813	−0.0108757
$682$	0	0
$683$	25.4164	0.972531	0.486266	−	0.873811i	$-0.338359\pi$
$683$	25.4164	0.972531	0.486266	+	0.873811i	$0.338359\pi$
$684$	0	0
$685$	67.0979	2.56368
$686$	0	0
$687$	17.0508	0.650529
$688$	0	0
$689$	4.26118	0.162338
$690$	0	0
$691$	17.5592	0.667985	0.333993	−	0.942576i	$-0.391604\pi$
$691$	17.5592	0.667985	0.333993	+	0.942576i	$0.391604\pi$
$692$	0	0
$693$	−3.18818	−0.121109
$694$	0	0
$695$	−14.0713	−0.533753
$696$	0	0
$697$	2.11843	0.0802411
$698$	0	0
$699$	2.24231	0.0848121
$700$	0	0
$701$	−8.15151	−0.307878	−0.153939	−	0.988080i	$-0.549196\pi$
$701$	−8.15151	−0.307878	−0.153939	+	0.988080i	$0.549196\pi$
$702$	0	0
$703$	−34.2122	−1.29034
$704$	0	0
$705$	−23.3119	−0.877978
$706$	0	0
$707$	−16.0635	−0.604131
$708$	0	0
$709$	6.77253	0.254348	0.127174	−	0.991880i	$-0.459409\pi$
$709$	6.77253	0.254348	0.127174	+	0.991880i	$0.459409\pi$
$710$	0	0
$711$	5.03225	0.188724
$712$	0	0
$713$	−3.29617	−0.123442
$714$	0	0
$715$	33.8136	1.26456
$716$	0	0
$717$	−3.40186	−0.127045
$718$	0	0
$719$	18.3219	0.683293	0.341646	−	0.939829i	$-0.389015\pi$
$719$	18.3219	0.683293	0.341646	+	0.939829i	$0.389015\pi$
$720$	0	0
$721$	2.50133	0.0931543
$722$	0	0
$723$	−17.0872	−0.635481
$724$	0	0
$725$	39.7467	1.47616
$726$	0	0
$727$	−13.7886	−0.511393	−0.255696	−	0.966757i	$-0.582305\pi$
$727$	−13.7886	−0.511393	−0.255696	+	0.966757i	$0.582305\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	5.27782	0.195207
$732$	0	0
$733$	−21.1474	−0.781098	−0.390549	−	0.920582i	$-0.627715\pi$
$733$	−21.1474	−0.781098	−0.390549	+	0.920582i	$0.627715\pi$
$734$	0	0
$735$	21.7837	0.803505
$736$	0	0
$737$	−24.3263	−0.896070
$738$	0	0
$739$	16.3081	0.599903	0.299951	−	0.953954i	$-0.403030\pi$
$739$	16.3081	0.599903	0.299951	+	0.953954i	$0.403030\pi$
$740$	0	0
$741$	8.40898	0.308911
$742$	0	0
$743$	−8.49786	−0.311756	−0.155878	−	0.987776i	$-0.549821\pi$
$743$	−8.49786	−0.311756	−0.155878	+	0.987776i	$0.549821\pi$
$744$	0	0
$745$	−14.8197	−0.542950
$746$	0	0
$747$	3.32413	0.121624
$748$	0	0
$749$	−4.78940	−0.175001
$750$	0	0
$751$	−4.21411	−0.153775	−0.0768876	−	0.997040i	$-0.524498\pi$
$751$	−4.21411	−0.153775	−0.0768876	+	0.997040i	$0.524498\pi$
$752$	0	0
$753$	30.1261	1.09786
$754$	0	0
$755$	52.0591	1.89463
$756$	0	0
$757$	−32.1650	−1.16906	−0.584529	−	0.811373i	$-0.698721\pi$
$757$	−32.1650	−1.16906	−0.584529	+	0.811373i	$0.698721\pi$
$758$	0	0
$759$	5.81857	0.211201
$760$	0	0
$761$	−19.2414	−0.697500	−0.348750	−	0.937216i	$-0.613394\pi$
$761$	−19.2414	−0.697500	−0.348750	+	0.937216i	$0.613394\pi$
$762$	0	0
$763$	7.50710	0.271775
$764$	0	0
$765$	2.38209	0.0861247
$766$	0	0
$767$	−2.33638	−0.0843618
$768$	0	0
$769$	19.1211	0.689525	0.344762	−	0.938690i	$-0.387959\pi$
$769$	19.1211	0.689525	0.344762	+	0.938690i	$0.387959\pi$
$770$	0	0
$771$	−12.5832	−0.453173
$772$	0	0
$773$	−18.6237	−0.669849	−0.334924	−	0.942245i	$-0.608711\pi$
$773$	−18.6237	−0.669849	−0.334924	+	0.942245i	$0.608711\pi$
$774$	0	0
$775$	14.7619	0.530262
$776$	0	0
$777$	−11.5451	−0.414179
$778$	0	0
$779$	−9.35027	−0.335008
$780$	0	0
$781$	36.1676	1.29418
$782$	0	0
$783$	4.94801	0.176827
$784$	0	0
$785$	−16.2483	−0.579928
$786$	0	0
$787$	47.1489	1.68068	0.840339	−	0.542062i	$-0.182356\pi$
$787$	47.1489	1.68068	0.840339	+	0.542062i	$0.182356\pi$
$788$	0	0
$789$	−30.6752	−1.09206
$790$	0	0
$791$	−5.03175	−0.178909
$792$	0	0
$793$	28.5678	1.01447
$794$	0	0
$795$	−5.32790	−0.188961
$796$	0	0
$797$	10.7918	0.382267	0.191133	−	0.981564i	$-0.438784\pi$
$797$	10.7918	0.382267	0.191133	+	0.981564i	$0.438784\pi$
$798$	0	0
$799$	−4.26085	−0.150738
$800$	0	0
$801$	−7.58863	−0.268131
$802$	0	0
$803$	−34.6656	−1.22332
$804$	0	0
$805$	6.36394	0.224300
$806$	0	0
$807$	−19.8127	−0.697441
$808$	0	0
$809$	19.5806	0.688416	0.344208	−	0.938893i	$-0.388147\pi$
$809$	19.5806	0.688416	0.344208	+	0.938893i	$0.388147\pi$
$810$	0	0
$811$	7.14317	0.250831	0.125415	−	0.992104i	$-0.459974\pi$
$811$	7.14317	0.250831	0.125415	+	0.992104i	$0.459974\pi$
$812$	0	0
$813$	−4.89053	−0.171518
$814$	0	0
$815$	−47.7269	−1.67180
$816$	0	0
$817$	−23.2952	−0.814994
$818$	0	0
$819$	2.83766	0.0991560
$820$	0	0
$821$	20.0758	0.700649	0.350325	−	0.936628i	$-0.386071\pi$
$821$	20.0758	0.700649	0.350325	+	0.936628i	$0.386071\pi$
$822$	0	0
$823$	−37.7764	−1.31680	−0.658401	−	0.752667i	$-0.728767\pi$
$823$	−37.7764	−1.31680	−0.658401	+	0.752667i	$0.728767\pi$
$824$	0	0
$825$	−26.0584	−0.907237
$826$	0	0
$827$	16.1818	0.562696	0.281348	−	0.959606i	$-0.409218\pi$
$827$	16.1818	0.562696	0.281348	+	0.959606i	$0.409218\pi$
$828$	0	0
$829$	−3.75821	−0.130528	−0.0652641	−	0.997868i	$-0.520789\pi$
$829$	−3.75821	−0.130528	−0.0652641	+	0.997868i	$0.520789\pi$
$830$	0	0
$831$	4.15322	0.144074
$832$	0	0
$833$	3.98154	0.137952
$834$	0	0
$835$	89.1234	3.08424
$836$	0	0
$837$	1.83768	0.0635195
$838$	0	0
$839$	20.2261	0.698284	0.349142	−	0.937070i	$-0.386473\pi$
$839$	20.2261	0.698284	0.349142	+	0.937070i	$0.386473\pi$
$840$	0	0
$841$	−4.51722	−0.155766
$842$	0	0
$843$	31.7832	1.09467
$844$	0	0
$845$	16.8354	0.579156
$846$	0	0
$847$	−0.468474	−0.0160970
$848$	0	0
$849$	11.6889	0.401161
$850$	0	0
$851$	21.0703	0.722282
$852$	0	0
$853$	−11.1132	−0.380508	−0.190254	−	0.981735i	$-0.560931\pi$
$853$	−11.1132	−0.380508	−0.190254	+	0.981735i	$0.560931\pi$
$854$	0	0
$855$	−10.5140	−0.359572
$856$	0	0
$857$	−32.1874	−1.09950	−0.549750	−	0.835329i	$-0.685277\pi$
$857$	−32.1874	−1.09950	−0.549750	+	0.835329i	$0.685277\pi$
$858$	0	0
$859$	−9.71379	−0.331430	−0.165715	−	0.986174i	$-0.552993\pi$
$859$	−9.71379	−0.331430	−0.165715	+	0.986174i	$0.552993\pi$
$860$	0	0
$861$	−3.15531	−0.107533
$862$	0	0
$863$	−12.5958	−0.428765	−0.214383	−	0.976750i	$-0.568774\pi$
$863$	−12.5958	−0.428765	−0.214383	+	0.976750i	$0.568774\pi$
$864$	0	0
$865$	54.3749	1.84880
$866$	0	0
$867$	−16.5646	−0.562564
$868$	0	0
$869$	−16.3245	−0.553769
$870$	0	0
$871$	21.6518	0.733642
$872$	0	0
$873$	−2.87712	−0.0973757
$874$	0	0
$875$	−10.7607	−0.363779
$876$	0	0
$877$	43.7071	1.47588	0.737942	−	0.674865i	$-0.235798\pi$
$877$	43.7071	1.47588	0.737942	+	0.674865i	$0.235798\pi$
$878$	0	0
$879$	29.6797	1.00107
$880$	0	0
$881$	−14.6184	−0.492507	−0.246253	−	0.969205i	$-0.579200\pi$
$881$	−14.6184	−0.492507	−0.246253	+	0.969205i	$0.579200\pi$
$882$	0	0
$883$	−14.2901	−0.480899	−0.240450	−	0.970662i	$-0.577295\pi$
$883$	−14.2901	−0.480899	−0.240450	+	0.970662i	$0.577295\pi$
$884$	0	0
$885$	2.92126	0.0981970
$886$	0	0
$887$	−29.5310	−0.991554	−0.495777	−	0.868450i	$-0.665117\pi$
$887$	−29.5310	−0.991554	−0.495777	+	0.868450i	$0.665117\pi$
$888$	0	0
$889$	−17.3440	−0.581699
$890$	0	0
$891$	−3.24397	−0.108677
$892$	0	0
$893$	18.8065	0.629334
$894$	0	0
$895$	79.2274	2.64828
$896$	0	0
$897$	−5.17885	−0.172917
$898$	0	0
$899$	9.09285	0.303264
$900$	0	0
$901$	−0.973810	−0.0324423
$902$	0	0
$903$	−7.86110	−0.261601
$904$	0	0
$905$	73.4249	2.44073
$906$	0	0
$907$	7.87205	0.261387	0.130694	−	0.991423i	$-0.458280\pi$
$907$	7.87205	0.261387	0.130694	+	0.991423i	$0.458280\pi$
$908$	0	0
$909$	−16.3446	−0.542116
$910$	0	0
$911$	−25.6372	−0.849398	−0.424699	−	0.905335i	$-0.639620\pi$
$911$	−25.6372	−0.849398	−0.424699	+	0.905335i	$0.639620\pi$
$912$	0	0
$913$	−10.7834	−0.356877
$914$	0	0
$915$	−35.7193	−1.18084
$916$	0	0
$917$	1.67851	0.0554293
$918$	0	0
$919$	47.6491	1.57180	0.785900	−	0.618354i	$-0.212200\pi$
$919$	47.6491	1.57180	0.785900	+	0.618354i	$0.212200\pi$
$920$	0	0
$921$	8.25970	0.272166
$922$	0	0
$923$	−32.1912	−1.05959
$924$	0	0
$925$	−94.3633	−3.10265
$926$	0	0
$927$	2.54509	0.0835918
$928$	0	0
$929$	−40.1854	−1.31844	−0.659221	−	0.751949i	$-0.729114\pi$
$929$	−40.1854	−1.31844	−0.659221	+	0.751949i	$0.729114\pi$
$930$	0	0
$931$	−17.5736	−0.575952
$932$	0	0
$933$	−22.6765	−0.742397
$934$	0	0
$935$	−7.72743	−0.252714
$936$	0	0
$937$	0.331054	0.0108151	0.00540753	−	0.999985i	$-0.498279\pi$
$937$	0.331054	0.0108151	0.00540753	+	0.999985i	$0.498279\pi$
$938$	0	0
$939$	18.6755	0.609451
$940$	0	0
$941$	44.4168	1.44795	0.723973	−	0.689828i	$-0.242314\pi$
$941$	44.4168	1.44795	0.723973	+	0.689828i	$0.242314\pi$
$942$	0	0
$943$	5.75857	0.187525
$944$	0	0
$945$	−3.54803	−0.115417
$946$	0	0
$947$	52.5325	1.70708	0.853538	−	0.521030i	$-0.174452\pi$
$947$	52.5325	1.70708	0.853538	+	0.521030i	$0.174452\pi$
$948$	0	0
$949$	30.8544	1.00158
$950$	0	0
$951$	−10.1281	−0.328425
$952$	0	0
$953$	32.3105	1.04664	0.523320	−	0.852136i	$-0.324693\pi$
$953$	32.3105	1.04664	0.523320	+	0.852136i	$0.324693\pi$
$954$	0	0
$955$	−18.1965	−0.588825
$956$	0	0
$957$	−16.0512	−0.518861
$958$	0	0
$959$	−18.2665	−0.589856
$960$	0	0
$961$	−27.6229	−0.891062
$962$	0	0
$963$	−4.87320	−0.157037
$964$	0	0
$965$	−6.42274	−0.206755
$966$	0	0
$967$	28.4425	0.914648	0.457324	−	0.889300i	$-0.348808\pi$
$967$	28.4425	0.914648	0.457324	+	0.889300i	$0.348808\pi$
$968$	0	0
$969$	−1.92171	−0.0617342
$970$	0	0
$971$	−29.2913	−0.940004	−0.470002	−	0.882665i	$-0.655747\pi$
$971$	−29.2913	−0.940004	−0.470002	+	0.882665i	$0.655747\pi$
$972$	0	0
$973$	3.83071	0.122807
$974$	0	0
$975$	23.1934	0.742785
$976$	0	0
$977$	−22.6562	−0.724835	−0.362417	−	0.932016i	$-0.618048\pi$
$977$	−22.6562	−0.724835	−0.362417	+	0.932016i	$0.618048\pi$
$978$	0	0
$979$	24.6173	0.786772
$980$	0	0
$981$	7.63845	0.243877
$982$	0	0
$983$	−16.2148	−0.517173	−0.258587	−	0.965988i	$-0.583257\pi$
$983$	−16.2148	−0.517173	−0.258587	+	0.965988i	$0.583257\pi$
$984$	0	0
$985$	−11.4791	−0.365755
$986$	0	0
$987$	6.34636	0.202007
$988$	0	0
$989$	14.3468	0.456203
$990$	0	0
$991$	49.9491	1.58669	0.793343	−	0.608774i	$-0.208339\pi$
$991$	49.9491	1.58669	0.793343	+	0.608774i	$0.208339\pi$
$992$	0	0
$993$	−26.0075	−0.825324
$994$	0	0
$995$	43.5074	1.37928
$996$	0	0
$997$	−58.5213	−1.85339	−0.926694	−	0.375818i	$-0.877362\pi$
$997$	−58.5213	−1.85339	−0.926694	+	0.375818i	$0.877362\pi$
$998$	0	0
$999$	−11.7471	−0.371663

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.g.1.3	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.g.1.3	✓	15	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6036.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.61011 q^{5} +0.982804 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.61011 q^{5} +0.982804 q^{7} +1.00000 q^{9} -3.24397 q^{11} +2.88731 q^{13} -3.61011 q^{15} -0.659840 q^{17} +2.91239 q^{19} +0.982804 q^{21} -1.79366 q^{23} +8.03288 q^{25} +1.00000 q^{27} +4.94801 q^{29} +1.83768 q^{31} -3.24397 q^{33} -3.54803 q^{35} -11.7471 q^{37} +2.88731 q^{39} -3.21052 q^{41} -7.99865 q^{43} -3.61011 q^{45} +6.45741 q^{47} -6.03410 q^{49} -0.659840 q^{51} +1.47583 q^{53} +11.7111 q^{55} +2.91239 q^{57} -0.809188 q^{59} +9.89425 q^{61} +0.982804 q^{63} -10.4235 q^{65} +7.49893 q^{67} -1.79366 q^{69} -11.1492 q^{71} +10.6862 q^{73} +8.03288 q^{75} -3.18818 q^{77} +5.03225 q^{79} +1.00000 q^{81} +3.32413 q^{83} +2.38209 q^{85} +4.94801 q^{87} -7.58863 q^{89} +2.83766 q^{91} +1.83768 q^{93} -10.5140 q^{95} -2.87712 q^{97} -3.24397 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) \) 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9	\( 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9} - 5 q^{11} - 14 q^{13} - 11 q^{15} - 5 q^{17} + 3 q^{19} - 4 q^{21} - 32 q^{23} + 2 q^{25} + 15 q^{27} - 23 q^{29} - 13 q^{31} - 5 q^{33} - 16 q^{35} - 10 q^{37} - 14 q^{39} - 14 q^{41} + 4 q^{43} - 11 q^{45} - 20 q^{47} - 9 q^{49} - 5 q^{51} - 30 q^{53} - 10 q^{55} + 3 q^{57} - 14 q^{59} - 38 q^{61} - 4 q^{63} - 24 q^{65} - 8 q^{67} - 32 q^{69} - 41 q^{71} - 19 q^{73} + 2 q^{75} - 39 q^{77} - 27 q^{79} + 15 q^{81} - 17 q^{83} - 6 q^{85} - 23 q^{87} - 23 q^{89} + 4 q^{91} - 13 q^{93} - 30 q^{95} - 18 q^{97} - 5 q^{99}+O(q^{100}) \) 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9 - 5 * q^11 - 14 * q^13 - 11 * q^15 - 5 * q^17 + 3 * q^19 - 4 * q^21 - 32 * q^23 + 2 * q^25 + 15 * q^27 - 23 * q^29 - 13 * q^31 - 5 * q^33 - 16 * q^35 - 10 * q^37 - 14 * q^39 - 14 * q^41 + 4 * q^43 - 11 * q^45 - 20 * q^47 - 9 * q^49 - 5 * q^51 - 30 * q^53 - 10 * q^55 + 3 * q^57 - 14 * q^59 - 38 * q^61 - 4 * q^63 - 24 * q^65 - 8 * q^67 - 32 * q^69 - 41 * q^71 - 19 * q^73 + 2 * q^75 - 39 * q^77 - 27 * q^79 + 15 * q^81 - 17 * q^83 - 6 * q^85 - 23 * q^87 - 23 * q^89 + 4 * q^91 - 13 * q^93 - 30 * q^95 - 18 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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